The Influence of Grid Shape and Asynchronicity
on Cellular Evolutionary Algorithms
Bernabé Dorronsoro
Enrique Alba
Mario Giacobini
Marco Tomassini
Central Computing Services
University of Málaga
Málaga, Spain
[email protected]
Computer Science Dpt.
University of Málaga
Málaga, Spain
[email protected]
Information Systems Dpt.
University of Lausanne
Lausanne, Switzerland
[email protected]
Information Systems Dpt.
University of Lausanne
Lausanne, Switzerland
[email protected]
Abstract— In this paper we study cellular evolutionary algorithms, a kind of decentralized heuristics, and the importance of
the induced exploration/exploitation balance on different problems. It is shown that, by choosing synchronous or asynchronous
update policies, the selection pressure, and thus the exploration/exploitation tradeoff, can be influenced directly, without
using additional ad hoc parameters. Synchronous algorithms
of different neighborhood-to-topology ratio, and asynchronous
update policies are applied to a set of benchmark problems. Our
conclusions show that the update methods of the asynchronous
versions, as well as the ratio of the decentralized algorithm, have
a marked influence on its convergence and on its accuracy.
I. I NTRODUCTION
This paper focusses on the class of algorithms called cellular
evolutionary algorithms (cEAs). We here present the canonical algorithm and suggest several variants targeted to solve
complex problems accurately with a minimum customization
effort. These techniques, also called diffusion or fine-grained
models, have been popularized, among others, by early work
of Gorges-Schleuter [1] and Manderick and Spiessen [2]. The
basic idea behind their behavior is to add some structure to
the population of tentative solutions. The pursued effect is
to improve on the diversity and exploration capabilities of the
algorithm while still admitting an easy combination with local
search and other search techniques to improve exploitation.
The above mentioned structured models are based on a
spatially distributed population in which genetic operations
may only take place in a small neighborhood of each individual. Usually, individuals are arranged on a regular grid
of dimensions d = 1, 2, or 3. Cellular EAs are a kind
of decentralized EA model [3]. They are not just a parallel
implementation of an EA; in fact, although parallelism could
be used to speed up the search, we do not address parallel
implementations in this work. However, it is worth remarking
that, although SIMD (single instruction stream - multiple data
streams) machine implementations were popular a decade ago,
this is no longer true.
Although fundamental theory is still an open research line
for cEAs, they have been empirically reported as being useful
in maintaining diversity and promoting slow diffusion of
solutions through the grid. Part of their behavior is due to
a lower selection pressure compared to that of panmictic EAs
(here panmictic means that any individual may mate with any
other in the population). The influence of the selection method,
neighborhood, and grid topology on the efficiency of cEAs in
comparison to other EAs have all been investigated in detail
in [4], [5], [6], [7], and tested on different applications in the
fields of combinatorial and numerical optimization.
Cellular EAs can be seen as stochastic cellular automata
(CAs) [8], [9] where the cardinality of the set of states is equal
to the number of points in the search space. CAs, as well as
cEAs, usually assume a synchronous or “parallel” update policy, in which all the cells are formally updated simultaneously.
However, this is not the only option available. Indeed, several
works on asynchronous CAs have shown that sequential update policies have a marked effect on their dynamics (see e.g.
[10], [11]). In addition, the shape of the structure in which
individuals evolve has a deep impact on the performance of the
cEA. The algorithm admits a specially easy modulation of its
shape that can sharpen the exploration or the exploitation capabilities of the canonical technique, as shown in [7]. Thus, it
is interesting to investigate asynchronous cEAs and non-square
shaped cEAs, in order to analyze their problem solving capabilities, which is the subject of this paper.
This work is organized as follows. The next section contains
some background on synchronous and asynchronous cEAs.
In Section III we discuss the ability of cEAs for changing
their behavior depending on the population shape. We briefly
study in Section IV the theoretical behavior of the proposed
algorithms. In sections V and VI we deal with a set of
benchmark problems with the goal of illustrating the actual
computational power of these algorithms for optimization.
Finally, section VII offers our conclusions, as well as some
comments on future work.
II. S YNCHRONOUS AND A SYNCHRONOUS C EA S
A cEA starts with the cells (individuals) in a random state
and proceeds by successively updating them using evolutionary operators, until a termination condition is met. Updating
a cell in a cellular EA means selecting two parents in the
individual’s neighborhood (including the individual itself),
applying genetic operators to them, and finally replacing the
individual if an offspring has a better fitness (or using another
replacement policy). The following pseudo-code is a high-level
description of the algorithm for a two-dimensional grid of size
HEIGHT×WIDTH and for formally simultaneous update of all
the cells. In fact, true simultaneous update could be performed
in a parallel computer, but usually this is simulated by using a
sequential machine and a second array to hold the updated cells.
Algorithm 1 Pseudocode of a synchronous cGA
1: proc Steps Up(cga)
//Algorithm parameters in ‘cga’
2: while not Stop Condition() do
3: for x ← 1 to WIDTH do
4:
for y ← 1 to HEIGHT do
5:
n list←Get Neighborhood(cga,position(x,y));
6:
parents←Local Select(n list);
7:
aux indiv←Recombination(cga.Pc,parents);
8:
aux indiv←Mutation(cga.Pm,aux indiv);
9:
aux indiv←Local Search(cga.Pl,aux indiv);
10:
Evaluate Fitness(aux indiv);
11:
Insert If Better(position(x,y),aux indiv,cga,aux pop);
12:
end for
13: end for
14: cga.pop←aux pop;
15: Update Statistics(cga);
16: end while
17: end proc Steps Up;
Cells can be updated synchronously or asynchronously. In
synchronous (parallel) update all the cells change their states
simultaneously, while in asynchronous, or sequential update,
cells are updated one at a time in some order. There exist many
ways for sequentially updating the cells of a cEA (an excellent
discussion of asynchronous update in cellular automata, which
are essentially the same system as a cEA, is available in [10]).
We consider four asynchronous update methods: fixed line
sweep (LS), fixed random sweep (FRS), new random sweep
(NRS), and uniform choice (UC).
•
•
•
•
In fixed line sweep (LS), the simplest method, the n grid
cells are updated sequentially (1, 2, . . . , n) line after line.
In fixed random sweep (FRS), the next cell to be updated is chosen with uniform probability without replacement; this will produce a certain update sequence
p
(cj1 , ck2 , . . . , cm
n ), where cq means that cell number p is
updated at time q and (j, k, . . . , m) is a permutation of
the n cells. The same permutation is then used for all
update cycles.
The new random sweep method (NRS) works like FRS,
except that a new random cell permutation is used for
each sweep through the array.
In uniform choice (UC), the next cell to be updated
is chosen at random with uniform probability and with
replacement. This corresponds to a binomial distribution
for the update probability.
A time step is defined as updating n times sequentially,
which corresponds to updating all the n cells in the grid for
LS, FRS, and NRS, and possibly less than n different cells in
the UC method, since some cells might be updated more than
once in a single time step. It should be noted that, with the
exception of fixed line sweep, the other asynchronous updating
policies are stochastic, representing an additional source of
non-determinism besides that of the genetic operators.
III. N EW C EA VARIANTS BASED ON A M ODIFIED R ATIO
After explaining our basic algorithm and the asynchronous
variants in the previous section, we now proceed to characterize the population grid itself. For this goal, we use the “radius” definition given in [7], which is refined from the seminal
one appeared in [6] to account for non square grids. The grid
is considered to have a radius equal to the dispersion of n∗
points in a circle centered in (x, y) (Eq. 1). This definition
always assigns different numerical values to different grids.
rad =
rP
(xi − x)2 +
n∗
x=
Pn∗
i=1
n∗
xi
P
(yi − y)2
, y=
Pn∗
i=1
n∗
,
(1)
yi
Although it is called a “radius”, rad measures the dispersion of n∗ patterns. Other possible measures for symmetrical
topologies would allocate the same numeric value to different
topologies (which is undesirable). Two examples are the radius
of a circle surrounding a rectangle containing the topology,
or an asymmetry coefficient. The definition (1) does not only
characterize the grid shape but it also can provide a radius
value for the neighborhood. As proposed in [6], the gridto-neighborhood relationship can be quantified by the ratio
between their radii (Eq. 2).
ratiocEA =
radN eighborhood
radT opology
(2)
When solving a given problem with a constant number
of individuals (n = n∗ , for making fair comparisons) the
topology radius will increase as the grid gets thinner (Fig. 1b).
Since the neighborhood is kept constant in size and shape
throughout this paper (we always use linear5 –L5–, Fig. 1a),
the ratio becomes smaller as the grid gets thinner.
2
(a)
(b)
Fig. 1. (a) Radius of neighborhood L5. (b) 5×5 = 25 and 3×8 ≈ 25 grids;
approximately equal number of individuals with two different ratios
After presenting this characterization of the radius and
topology by means of a ratio value, the main question still
remains to be posed: what is the importance of such a ratio
measure? The answer comes when we get into the actual
meaning of the ratio, i.e., reducing the ratio means reducing the
global selection intensity on the population (see next section),
thus promoting exploration. This is expected to allow for a
higher diversity in the genotype that could improve the results
in difficult problems (like in multimodal or epistatic tasks). On
the other hand, the search performed inside each neighborhood
is guiding the exploitation of the algorithm. We study in this
IV. S ELECTION P RESSURE , G RID S HAPE , AND T IME
Selection pressure is related to the concept of takeover
time, which is the time it takes for a single best individual
to colonize the whole population with copies of itself under
the effects of selection only [15]. Shorter takeover times mean
a stronger selection.
We plot in figures 2, 3, and 4 the takeover times for some
synchronous and asynchronous cEAs. For that, we only maintain selection active, and we also use binary tournament. Nor
mutation neither crossover are applied during the evolution.
Initially, we place only one optimal individual in a random
location of the grid, and we let it be selected and copied until
it takes over all the grid positions. The axes of the graphics
represent the proportion of the best individual in the population
(X axis), and the number of generations or time steps (Y axis).
Algorithms with similar ratio show a similar selection
pressure, as stated in [5]. In Fig. 2 we plot such a similar
behavior for two algorithms with different neighborhood and
population radii, but having two similar ratio values. The cases
plotted are those using a L5 neighborhood with a 32 × 32
population, and a compact21 —C21— neighborhood with a
population of 64 × 64 individuals. In the C21 neighborhood
a central cell is surrounded by two cells in all directions,
including the diagonals, and the four corner cells are cut out.
Hence, it may be interesting to see how the shape of the
grid influences the search of the algorithm. The selection
pressure for different cEAs using L5 neighborhood and 6
possible grid shapes are plotted in Fig. 3 for a population
of 1024 individuals. Note that the selection pressure induced
in synchronous rectangular grids falls under the curve for a
synchronous square grid (32 × 32 population), which means
that thinner grids favor a more explorative style of search.
If we now keep the shape of the grid constant (say a
square) but we allow the cell update mode to change, we
observe a similar effect on the selection pressure. In fact, it is
found that the global selection pressure induced by the various
asynchronous policies falls between the low synchronous limit
and the high panmictic bound (see Fig. 4 and [16]). Thus,
by varying the update policies it is possible to influence the
explorative or exploitative character of the search.
Best Individual Proportion
N. of Generations
Fig. 2. Growth curves of the best individual for two cEAs with different
neighborhood and population shapes, but similar ratio values. The graph
represents the proportion of population consisting of best individual as a
function of time
1
0.9
Best Individual Proportion
paper how the ratio affects the search efficiency over a variety
of domains. Changing the ratio during the search is a unique
feature of cEAs that can be used to shift from exploration to
exploitation at a minimum complexity without introducing just
another new algorithm family in the literature.
Many other techniques for managing the exploration/exploitation tradeoff are possible. Among them, it is worth to make
a special mention to heterogeneous EAs [12] [13], in which
algorithms with different features run in separate subpopulations and collaborate in order to avoid premature convergence.
A different alternative is using Memetic Algorithms [14], in
which local search is combined with the genetic operators in
order to promote local exploitation.
0.8
ratio 0.003 (1x1024 pop.)
ratio 0.006 (2x512 pop.)
ratio 0.012 (4x256 pop.)
ratio 0.024 (8x128 pop.)
ratio 0.047 (16x256 pop.)
ratio 0.069 (32x32 pop.)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
1
10
2
10
N. of Generations
3
10
Fig. 3. Takeover times with tournament selection using a L5 neighborhood
in populations of 1024 individuals with different grid shapes. Mean values
over 100 runs. The graph represents the proportion of population consisting
of best individual as a function of time. Horizontal axis is in logarithmic scale
Fig. 4. Takeover times with tournament selection using a L5 neighborhood
in a 32 × 32 grid. Mean values over 100 runs. The graph represents the
proportion of population consisting of best individual as a function of time
V. T EST S UITE
B. Multimodal Problem Generator (P-PEAKS)
In this section we present the set of problems chosen
for the experimental study. The benchmark is representative
because it contains many different interesting features found in
optimization, such as epistasis, multimodality, deceptiveness,
and problem generators. These are important ingredients in
any work trying to evaluate algorithmic approaches with the
objective of getting reliable results, as stated by Whitley et al.
in [17].
We experiment with the massively multimodal deceptive
problem (MMDP), and the multimodal problem generator
P-PEAKS, which are included in the set of problems studied
in [7]; next we will extend this basic two-problem benchmark
with error correcting code design (ECC), and maximum cut
of a graph (MAXCUT). The choice of this set of problems is
justified by both their difficulty and their application domains
(telecommunications, combinatorial optimization, etc.). This
gives us a high level of confidence in the results, although the
evaluation of conclusions will result more laborious than with
a smaller test suite.
The selected problems are explained in subsections V-A to
V-D. We include the explanations in this paper to make it
self-contained and to avoid the typical small lacks that could
preclude other researchers from reproducing the results.
A. Massively Multimodal Deceptive Problem (MMDP)
The MMDP is a problem that has been specifically designed
to be difficult for an EA [18]. It is made up of k deceptive subproblems (si ) of 6 bits each one, whose value depends on the
number of ones (unitation) a binary string has (see Fig. 5). It is
easy to see (graphic of Fig. 5) that these subfunctions have two
global maxima and a deceptive attractor in the middle point.
The P-PEAKS problem [19] is a multimodal problem generator. A problem generator is an easily parameterizable task
which has a tunable degree of epistasis, thus admitting to
derive instances with growing difficulty at will. Also, using a
problem generator removes the opportunity to arbitrarily handtune algorithms to a particular problem, therefore allowing a
larger fairness when comparing algorithms. With a problem
generator we evaluate our algorithms on a high number of
random problem instances, since a different instance is solved
each time the algorithm runs, then the predictive power of the
results for the problem class as a whole is increased.
The idea of P-PEAKS is to generate P random N -bit strings
that represent the location of P peaks in the search space. The
fitness value of a string is the number of bits the string has in
common with the nearest peak in that space, divided by N (as
shown in Eq. 4). By using a small/large number of peaks we
can get weakly/strongly epistatic problems. In this paper we
have used an instance of P = 100 peaks of length N = 100
bits each, which represents a medium/high epistasis level [7].
The maximum fitness value for this problem is 1.0.
fP −P EAKS (~x) =
1
max {N − HammingD(~x, P eaki )} (4)
N 1≤i≤p
C. Error Correcting Code Design Problem (ECC)
The ECC problem was presented in [20]. We will consider a
three-tuple (n, M, d), where n is the length of each codeword
(number of bits), M is the number of codewords, and d is the
minimum Hamming distance between any pair of codewords.
Our objective will be to find a code which has a value for
d as large as possible (reflecting greater tolerance to noise
and errors), given previously fixed values for n and M . The
problem we have studied is a simplified version of that in
[20]. In our case we search half of the codewords (M/2) that
will compose the code, and the other half is made up by the
complement of the codewords computed by the algorithm.
The fitness function to be maximized is:
fECC =
1
M
M
X
X
(5)
d−2
ij
i=1 j=1,i6=j
Fig. 5.
Basic deceptive bipolar function (si ) for MMDP
In MMDP each subproblem si contributes to the fitness
value according to its unitation (Fig. 5). The global optimum
has a value of k and it is attained when every subproblem
is composed of either zero or six ones. The number of local
optima is quite large (22k ), while there are only 2k global
solutions. Therefore, the degree of multimodality is regulated
by the k parameter. We use here a considerably large instance
of k = 40 subproblems. The instance we try to maximize for
solving the problem is shown in Eq. 3, and its maximum value
is equal to k.
fM M DP (~s) =
k
X
i=1
f itnesssi
(3)
where dij represents the Hamming distance between codewords i and j in the code C (made up of M codewords, each
of length n). We consider in the present paper an instance
where M = 24 and n = 12. The search space is of size
4096
, which is approximately 1087 . The optimum solu24
tion for M = 24 and n = 12 has a fitness value of 0.0674 [21].
D. Maximum Cut of a Graph (MAXCUT)
The MAXCUT problem looks for a partition of the set
of vertices (V ) of a weighted graph G = (V, E) into two
disjoint subsets V0 and V1 so that the sum of the weights of
the edges with one endpoint in V0 and the other one in V1 is
maximized. For encoding the problem we use a binary string
(x1 , x2 , . . . , xn ) of length n where each digit corresponds to
a vertex. If a digit is 1 then the corresponding vertex is in set
V1 ; if it is 0 then the corresponding vertex is in set V0 . The
function to be maximized [22] is:
n−1 X
n
X
fM AXCU T (~x) =
wij · xi · (1 − xj ) + xj · (1 − xi )
i=1 j=i+1
(6)
Note that wij contributes to the sum only if nodes i and j are
in different partitions. While one can generate different random
graph instances to test the algorithm, here we have used the
case “cut20.09”, with 20 vertices and a probability of 0.9 of
having an edge between any two randomly chosen vertices.
The maximum fitness value for this instance is 56.740064.
VI. E XPERIMENTAL A NALYSIS
Although a full-length study of the problems presented
in the previous section is beyond the scope of this work,
we present results comparing synchronous and asynchronous
cEAs, and also cEAs having different values of the ratio,
always with a constant neighborhood shape (L5). Note that
it is not our aim here to compare cEAs performance with
state-of-the-art algorithms and heuristics for combinatorial
and numerical optimization. To this end, we should at least
tune the parameters and include local search capabilities in
the algorithm, which is not the case. Thus, the results only
pertain to the relative performance of the different cEA update
methods and ratios among themselves.
Here we present the results of solving several problems
using JCell v1.0, our custom optimization program written
in Java, with three different static ratios, and the four asynchronous update modes previously described. The configuration of the algorithm is detailed in Table I, while the static
ratios used are shown in Table II.
TABLE I
steps to obtain the optimum value (if obtained) and the hit rate
(percentage of successful runs). Therefore, we are analyzing
the final distance to the optimum (especially interesting when
the optimum is not found), the effort of the algorithm, and its
expected efficacy, respectively.
TABLE III
MMDP PROBLEM WITH A MAXIMUM OF 1000 GENERATIONS
Algorithm
Square
Rectangular
Narrow
LS
FRS
NRS
UC
Avg. Solution (best=20)
19.813
19.824
19.842
19.518
19.601
19.536
19.615
Avg. Generations
214.18
236.10
299.67
343.52
209.94
152.93
295.72
Hit Rate
57%
58%
61%
23%
31%
28%
36%
TABLE IV
P-PEAKS PROBLEM WITH A MAXIMUM OF 100 GENERATIONS
Algorithm
Square
Rectangular
Narrow
LS
FRS
NRS
UC
Avg. Solution (best=1)
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Avg. Generations
51.84
50.43
53.94
34.75
38.39
38.78
40.14
Hit Rate
100%
100%
100%
100%
100%
100%
100%
TABLE V
ECC PROBLEM WITH A MAXIMUM OF 500 GENERATIONS
Algorithm
Square
Rectangular
Narrow
LS
FRS
NRS
UC
Avg. Solution (best=0.0674)
0.0670
0.0671
0.0673
0.0672
0.0672
0.0672
0.0671
Avg. Generations
93.92
93.35
104.16
79.66
82.38
79.46
87.27
Hit Rate
85%
88%
94%
89%
90%
89%
86%
PARAMETERIZATION USED IN THE ALGORITHM
Population Size
Selection of Parents
Recombination
Bit Mutation
Individual Length
Replacement
Stop Condition
400 Individuals
Binary Tournament + Binary Tournament
DPX, pc = 1.0
Bit-Flip, pm = 1/L
L
Rep if Better
Find Optimum or Max Number of Steps
TABLE II
S TUDIED RATIOS
Name
Square
Rectangular
Narrow
(shape of
(20 × 20
(10 × 40
(4 × 100
population)
individuals)
individuals)
individuals)
Value of ratio
0.11
0.075
0.031
We show in the following tables the results for the problems
before mentioned: MMDP (Table III), P-PEAKS (Table IV),
ECC (Table V), and MAXCUT (Table VI). We have performed
100 independent runs for any algorithm and for every problem
in the test-suite. In these tables we report the average of the
final best fitness on every run, the average number of time
TABLE VI
MAXCUT PROBLEM WITH A MAXIMUM OF 100 GENERATIONS
Algorithm
Square
Rectangular
Narrow
LS
FRS
NRS
UC
Avg. Solution (best=56.74)
56.74
56.74
56.74
56.74
56.74
56.74
56.74
Avg. Generations
11.26
11.03
11.88
9.46
9.69
9.55
9.58
Hit Rate
100%
100%
100%
100%
100%
100%
100%
From the inspection of these tables some conclusions can
be clearly drawn. First, the studied asynchronous algorithms
tend to need a smaller number of generations than the synchronous ones to locate an optimum, in general. Moreover,
referring to tables in Appendix where t-tests are reported
(character ‘+’ stands for significant values, while ‘-’ means
no significance), the reader will confirm that the differences
among asynchronous and synchronous algorithms are statistically significant, thus indicating that the asynchronous versions
perform more efficiently with respect to cEAs with different
static ratios. This result conforms to our study of Section IV
since asynchronous algorithms have a smaller takeover time.
There are however punctual exceptions, like in MMDP.
Conversely, if we pay attention to the success (hit) rate, it
can be concluded that the synchronous policies with changing
ratios outperform the asynchronous algorithms (except for
ECC): slightly in terms of the average final fitness, and clearly
in terms of probability of finding a solution (i.e., frequency of
optimum location).
Another interesting result is the fact that we can define two
classes of problems: those solved by any method to optimality
(100% hit rate) and those in which no 100% rate is achieved at
all. The former ones seem to be suitable for cEAs directly,
while the later ones need some help, e.g., by including local
search.
In order to summarize the large set of results and get some
useful conclusions we present a ranking with the best algorithms by following three different metrics: average best final
solution, average number of generations on success, and hit
rate. Table VII shows the three mentioned rankings. These
rankings have been computed by adding the position (from
better to worse: 1, 2, 3 ...) that algorithms are allocated for the
previous results presented from Table III to Table VI, according to the three criteria.
A PPENDIX
TABLE VII
R ANKING OF THE ALGORITHMS
1
2
2
4
5
5
7
Avg. Solution
Narrow
Rectangular
FRS
NRS
UC
LS
Square
4
9
9
10
11
11
12
1
2
3
4
5
6
7
Avg. Generations
NRS
8
LS
10
FRS
11
UC
16
Rectangular
19
Square
21
Narrow
27
cEAs are very efficient optimization techniques, that could
be further improved by being hybridized with local search
techniques [23]. The results on the test problems largely
confirm, with some small exceptions, that the solving abilities
using the various update/ratio modes are directly linked to
their induced selection pressures, showing that exploitation
plays an important role. It is clear that the role of exploration
might be more important on even harder problem instances,
but this aspect can be addressed in our algorithms by using
more explorative settings, as well as by using different cEA
strategies at different times during the search dynamically [24].
In summary, we have found asynchronous algorithms to
be numerically more efficient (faster) than synchronous ones
(with statistically significance) for P-PEAKS, ECC, and MAXCUT, but not for MMDP. On the other hand, synchronous
algorithms outperform asynchronous ones in terms of the hit
rate for our benchmark, which could be a very important
issue for many applications. As a future work, it would be
interesting to give a unified study of the different selection
intensities and their influence in the resolution of each problem
considered for the analysis.
1
2
2
4
5
5
5
Hit Rate
Narrow
Rectangular
FRS
NRS
Square
UC
LS
4
9
9
11
12
12
12
As we would expect after the previous comments, according
to the average final best fitness and hit rate criteria, synchronous algorithms with narrow and rectangular ratios are in
general more accurate than all the asynchronous ones for our
test problems, with a noticeable leading position for narrow
population grids. On the other hand, asynchronous versions
clearly outperform any of the synchronous algorithms in terms
of the average number of generations, with a trend towards
NRS as being the best ranked flavor of cEA for our test suite.
VII. C ONCLUSIONS
In the first part of this paper we have described several
asynchronous update policies for the population of a cEA,
followed by some ratio policies, all of them inducing a
different kind of search in the cEA. One can tune the selection
intensity of a cEA by choosing the update policy and/or
grid ratio without having to deal with additional numerical
parameter settings. This is a clear advantage of the algorithms
proposed in this study.
In the second part of the paper we have applied our extended
cEAs to a set of test problems. Although our goal has not
been that of obtaining solvers able to compete with state-ofthe-art specialized heuristics, the results point in that sense:
In this appendix we show a statistical comparison of the
studied algorithms by performing t-tests on the results of all
the algorithms. Tables VIII to XIII contain the values of our
statistical comparison in terms of the solutions found and the
number of generations. No tables are provided for those cases
in which the optimum is found every run (MAXCUT and
P-PEAKS). On the following tables, statistical significance
(5% level) is shown by using symbol ‘+’, while absence of
statistical significance is marked with ‘-’.
TABLE VIII
P - VALUES OF THE AVG . FITNESS FOR
MMDP
Algorithm
Square Rectangular Narrow LS FRS NRS UC
Square
•
−
− +
+
+ +
Rectangular
−
•
− +
+
+ +
Narrow
−
−
• +
+
+ +
LS
+
+
+ • −
− −
FRS
+
+
+ −
•
− −
NRS
+
+
+ − −
• −
UC
+
+
+ − −
− •
TABLE IX
P - VALUES OF THE GENERATIONS FOR
MMDP
Algorithm
Square Rectangular Narrow LS FRS NRS UC
Square
•
−
+ + −
− −
Rectangular
−
•
− − −
− −
Narrow
+
−
• − −
+ −
LS
+
−
− • −
+ −
FRS
−
−
− −
•
− −
NRS
−
−
+ + −
• +
UC
−
−
− − −
+ •
TABLE X
P - VALUES OF THE GENERATIONS FOR
P-PEAKS
Algorithm
Square Rectangular Narrow LS FRS NRS UC
Square
•
+
+ +
+
+ +
Rectangular
+
•
+ +
+
+ +
Narrow
+
+
• +
+
+ +
LS
+
+
+ •
+
+ +
FRS
+
+
+ +
•
− +
NRS
+
+
+ + −
• +
UC
+
+
+ +
+
+ •
TABLE XI
P - VALUES OF THE AVG . FITNESS FOR
ECC
Algorithm
Square Rectangular Narrow LS FRS NRS UC
Square
•
−
+ − −
− −
Rectangular
−
•
− − −
− −
Narrow
+
−
• − −
− +
LS
−
−
− • −
− −
FRS
−
−
− −
•
− −
NRS
−
−
− − −
• −
UC
−
−
+ − −
− •
TABLE XII
P - VALUES OF THE GENERATIONS FOR
ECC
Algorithm
Square Rectangular Narrow LS FRS NRS UC
Square
•
−
+ +
+
+ +
Rectangular
−
•
+ +
+
+ +
Narrow
+
+
• +
+
+ +
LS
+
+
+ • −
− +
FRS
+
+
+ −
•
− +
NRS
+
+
+ − −
• +
UC
+
+
+ +
+
+ •
TABLE XIII
P - VALUES OF THE GENERATIONS FOR
MAXCUT
Algorithm
Square Rectangular Narrow LS FRS NRS UC
Square
•
−
− +
+
+ +
Rectangular
−
•
− +
+
+ +
Narrow
−
−
• +
+
+ +
LS
+
+
+ • −
− −
FRS
+
+
+ −
•
− −
NRS
+
+
+ − −
• −
UC
+
+
+ − −
− •
ACKNOWLEDGEMENTS
This work has been partially funded by the Ministry of Science and Technology (MCYT) and Regional Development European Found (FEDER) under
contract TIC2002-04498-C05-02 (the TRACER project)
http://tracer.lcc.uma.es.
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